molecular orbital (MO) method, the other approach is more close to the valence bond (VB) treatment. For a comparison and .... as 3.066 eY, while a configuration interaction treatment, based on the .... Fd according to which we can find two.
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Recent Developments in the Method of Different Orbitals for Different Spins
P
f
.
MAR 1 6 1965
by Ruben Pauncz*
Quantum Theory Project Departments of Chemistry 8nd Physics Nuclear Sciences Building
;bz
University of Florida Gainesville, Florida 32603
Supported in part by the National Aeronautics--and Space Administration Research Gran
~
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~~~~
*On
leave from the Department of Chemistry, Technion, Israel Insti-
tute of Technology, Haifa.
ABSTRACT
The alternant molecular orbital method and the nonpaired spatial orbital method have been compared.
Some recent applications of the AM0 OTS PRICE(S) $
method have been reported.
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Hard copy (HC) Microfiche (MF)
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32
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INTRODUCTION
During recent years various extensions of the Hartree-Fock SCF method have been considered; for a general review we refer to the paper
'1 of Gwdin. J Among the extensions the alternant molecular orbital (AMO) method has gained considerable interest. The general characteristics
of the method and its applications
w
cussed by the autho
t o conjugated
systems have been dis-
quite recently. The aim of the present note is to
consider some recent applications of the method to non-alternant systems, as well as to contrast it with another approach, the Non Paired Spatial Orbital (NPSO) method, which seems to be even more powerful, when applied to small conjugated systeme. The AM0 method is closely related to the molecular orbital (MO) method, the other approach is more close to the valence bond (VB) treatment.
For a comparison and critical evaluation
of these two basic methods and their modifications, we refer to the /
paper of Slater given at the Symposi&L Alternant Molecular Orbital Method
4
The method has been suggested by Lbwdin ; for a general review of the method and its applications we refer to Ref. 2. Here we would like to mention a few points only which will be needed for the comparison with the NPSO method. Let us take a conjugated system consisting of 2 n centers and n? trons. We have to choose n
orbitals which form an orthonormal system.
Next we select n orthogonal two-dimensional subspaces spanned by the 0
elec-
-3
-
and $e? respectively (i= i' i I n each two-dimensional subspace we form two non-orthogonal
pair of orbitals, which will be denoted by 1,
...,n) .
linear combinations of 9:. and 1 1 '
I!J-
.i
$
in the following way:
The new set obtained consists of two subsets (aI;
i,),
I"1, ..A.
We have orthogonality within each subset, as well as between orbitals
f and i /iJVor
a i
pairs aI and H
IjJ; the only non-orthogonality will exist between the
I'
isl,. ..,n.
We emphasize this point because it has important consequences in the calculation of the energy and the applicability of the method to larger systems. The final wavefunction is built up by associating one set of orbitals
(/a&?
w i t h 5 spin functions, while the other set
(zI) with -@ spin
functions. As the determinantial function is not a proper eigenfunction of &we
have t o select the proper spin component.
This is conveniently
done by the use of the projection operator method, as proposed by Lbwdin5>/
-- . -4
-
The use of the projection operator in the construction of the wavefunction has important implications both with respect to the effective treatment of the correlation problem and with respect to the calculation of the energy; these aspects have been discussed extensively in Ref. 2. The matrix element of the Hamiltonian is given by the following expression
\
where Pf/means permutation of the spatial variables.
Because of the
orthogonality properties mentioned before, only certain permutations will lead to nonvanishing contributions,
so
possible integrals and their coefficients.
it is easy to classify the Eventually the integrals are
expressed in terms of integrals over the basic orbitals and certain
4,
functions of the parameters
So far we have not specified how to choose the basic orbitals and
the f3 two-dimensional subspaces. The best selection would be one which would lead t o the lowest energy value obtainable, subject to the limitation that the trial function is given in form (3) and (1).
This
problem is still unsolved; investigations in this direction are in progress.
A few guiding principles which help in choosing the orbitals are
the following: a)
If the parameters
8 reduce to zero, the functional form given 4
in Eq. (3) goes over to a MO description with doubly filled orbitals. This suggests the choice of the SCF orbitals as a natural starting point, because any other choice would mean a higher energy for this limiting point, but it should be observed that it is not necessarily the best choicku
-5-
b)
The criterion given in a) does not specify the orbitals uniquely.
An arbitrary unitary transformation would leave the energy value of the Hartree-Fock-Roothaan function invariant. T h i s transfomtion could be performed in such a way that the orbitals would be localized, see Edmiston r
and Ruedenberg's p a p a y The selection of the orbital pairs should be done such that the alternant orbital pairs would be well separated in space, so as t o yield a better correlation between electrons possessing
parallel spins. c)
It is desirable that the total wavefunction, which belongs to a
given irreducible representation of the symmetry group, should be represented as a spin projection of a single Slater determinant.
This is
imporant because it preserves the direct connection with the independent 1d.j particle model.
It leads also to
a considerable simplification in the
calculation of the energy.
In the prevfous applications
t o alternant conjugated systems, the
selection of the orbital pairs was related to those pairs of orbitals, which in the HUckel approximation have the following form:
Here the two s-tions
&/and +/J
h.{refer
-
to the two sfbsystems, LL
each of which has the properties that no two atoms in the subsystems are neighbors to each other. Recent Applications of the AMI Method There are two recent applications of the AM0 method to which ue would like to draw attention.
The first is concerned with applications
.
-6-
d
of the method to non-alternant systems, as azulene and fulvene
In these
cases we cannot select the pairs in the same way, as has been done in alternant molecules.
In the calculation the starting orbitals were formed
Y
from the eigenvectors of the 3verlap matrix, these are somewhat inferior to the SCF orbitals, but they
present considerable simplification in
the evaluation of electron repulsion integrals over the molecular orbitals.
The selection of the pairs have been made on the-basisof princi-
ples b) and c).
In the case of azulene, when using five different mixing
(J&?
parameters,
the energy improvement for the ground state was obtained
as 3.066 eY, while a configuration interaction treatment, based on the same orbitals, using
24 configurations yielded only 1.928 eV. Full
details of the calculations will be published in a separate note. The second application has been made to naphtalene, using five and comparing the result with a rather extensive
different parameters L
configuration interaction study which included 50 configurational
.
functions9
The AMD method using 5 nonlinear parameters turned out to be
more effective than the CI treatment with 49 linear parameters.
The
corresponding energy improvements for the ground state were as follows:
E
=
5.045 eV
( M ) ;3.393 (CI).
We cannot claim that these configuxations have been the 50 most important configurations, they have been chosen mostly on the basis of their energy values and it is well known that this criterion is unsufficient to determine their effectiveness in lowering the energy. Still it is rather impressive that the AM0 with a small number of parameters can surpass the limited CI treatment for a medium size system. The orbitals used in this calculations were SC-LCAO-MO orbitals.
It is interesting to compare
A
> 2
00 i r ---c
-7-
t h e r e s u l t s with t h a t r e l a t i n g t o the i s o e l e c t r o n i c non-alternant s y s t e m , azulene.
We s e e t h a t because of the p e c u l i a r geometrical s t r u c t u r e i n
t h e case of azulene, t h e energy improvement was smaller than i n naphtalene, but s t i l l s u b s t a n t i a l . The Non-Paired S p a t i a l O r b i t a l Method
Another important development during r e c e n t years has been t h e a p p l i c a t i o n of t h e Non-Paired S p a t i a l O r b i t a l (NPSO) method t o a number
of molecules.
The method has been suggested by Hirst and Linnet'tlO'and
t e s t e d i n a number of a p p l i c a t i o n s by L i n n e t t and coworkers.
For a com-
prehensive bibliography w e r e f e r t o the paper of H i r s t and L i n n e t t i n
'u!
this issue
The main o b j e c t i v e of t h e method is s i m i l a r t o t h a t of the AM0 method:
t o take i n t o account t h e c o r r e l a t i o n between e l e c t r o n s with
a n t i p a r a l l e l s p i n s and t o represent the t o t a l wavefunction by a f u n c t i o n a l
form which contains a small number of a d j u s t a b l e parameters.
The b a s i c
o r b i t a l s are two-center, semilocalized o r b i t a l s of t h e type considered by Mueller and E y r i n w a n d by Coulson and Fischer'3; hydrogen molecule.
and L i n n e t t
i n t h e case of the
Taking t h e benzene case, as an example (Empedocles
14) we have t h e following s p a t i a l o r b i t a l s .
The s p a t i a l o r b i t a l s a r e combined w i t h appropriate s p i n functions and t h e
/
Q5,/: t o t a l wave function has t h e following form
-8
q=
7
(oc!f-i-fi-~p
-
!2 ' - 0 i x y
1
7 21
f-
*33ici n / 2
-pp ;.yap)/G
/? -'
The f u n c t i o n a l form i s more general than t h a t used by t h e AM0 method i n
two respects: a) The s p a t i a l p a r t is a sum of two products i n s t e a d of the s i n g l e product occurring i n Eq. (3); b ) The l i n e a r combination of two independent s p i n p r o j e c t i o n s have been used.
The nonlinear parameter k
w i l l determine t h e shape of the basic two c e n t e r o r b i t a l s ; t h e l i n e a r parameter g i v e s t h e r e l a t i v e weight of the two s p i n functions
z",~ and I
$7
J'
The b e s t value of k f o r the ground s t a t e w a s equal t o 3.69.
The
r e s u l t obtained w a s very good, i t surpassed t h e corresponding AM0 t r e a t ment with two parameters.
The same r e s u l t has been obtained i n t h e case
of t h e a l l y l ions and r a d i c a y g t h e NPSO usually halves t h e d i s t a n c e between t h e AM0 r e s u l t and t h a t of a f u l l configuration i n t e r a c t i o n t r e a t ment.
The reason f o r the b e t t e r r e s u l t seems t o l i e i n t h e more general
f u n c t i o n a l form.
From the two aspects considered before, the f i r s t one
i s probably t h e more important.
Preliminary s t u d i e s i n t h e AM0 treatment
of the benzene case indicate t h a t the admission of other s p i n functions
*
leads t o an improvement which i s still smaller than the one obtained by t h e use of two d i f f e r e n t i n o t h e r systems too.
parameters.
This question has t o be s t u d i e d
The r e s u l t s of t h e NPSO method seem t o open a new p o s s i b i l i t y i n d e s c r i b i n g and i n t e r p r e t i n g t h e s t r u c t u r e of rnolecules1?/
The approach i s
c l o s e l y r e l a t e d t o t h e valence-bond method, b u t i t d i f f e r s from it i n ,
important respects.
One is t h e u s e of two c e n t e r bond o r b i t a l s instead
of t h e atomic ones, t h e second difference l i e s i n t h e use of a d i f f e r e n t
.i
-9-
spin coupling scheme.
The use of the best spin function, as Linnett and
caworkers point out, deserves further careful study. The method suffers a serious drawback from the computational point of view: we have to include all the permutations in the energy expression (like Eq. (4)), as we have no orthogonality properties among the spatial orbitals.
This will limit the applicability of the method to small
systems. This is the same drawback that prevented the valence bond method from nonempirical calculations of large systems. suggested by McWeen&!!(the
The remedy
introduction of orthogonalized atomic orbi-
tals instead of the original ones, does not look promising in thecase of NPSO, but it may be worth trying.
Another possibility would be the use
F d according to which we can find two
of the pairing principle (Ltfwdin
transformations which will replace the orbitals associated with CI or
f3
spinfunctions in the unprojected determinant in such a way that the new functions will exhibit rhe same properties, as expressed in Eq. (2). This would lead to simplifications in the calculation of the matrix elements, but it could not be used in such a natural way as in the case of the AMD method. In contrast to the NPSO method, the AM0 is better adapted to the treating of large conjugated systems. In fact, even an infinite linear 20,
chain can be satisfactorily treated by the use of the AM0 method,. 2 1,’ Applications to two dimensional systems are in progress 1
’
Conclusion In conclusion we can say that the use of different spatial orbitals in connection with different spinfunctions has showed that a suitably chosen functional form with a small number of non-linear variational
..
-10-
parameters can successfully approximate a configuration interaction wavefunction with a high number of linear parameters.
This is an important
achievement because for larger systems the full CI treatment becomes unmaoageable.
At the same time the method gave new problems to solve as the search for the best spinfunctlons. From the two methods considered in this note the NPSO gives better results for small systems, but the AM0 has wider possibilities for applications.
.
..
. ...
,
;:..
-...
.;
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-
-
I..
.
a
,
C
References 1.
P, 0. Lkdin, Molecular Orbitals in Chemistry, Physics, and Biology. A Tribute to R. S. Mulliken, Academic Press, New York, 1964. p. 37.
2.
R. Pauncz, Molecular Orbitals in Chemistry, Physics, and Biology. A Tribute to R. S. Mulliken, Academic Press, Kew York, 1364. p. 433.
3.
J. C. Slater, J. Chem. Phys., 00, 000 (this issue).
4.
P. 0. Lhdin, Symposium on Molecular Physics, Nikko, Japan, 1954, P. 599; PhP. Rev. pz, 1509 (1955).
5.
P. 0. Lbdin, Collolue inter. centre natl. recherche sci. Paris 82,
6.
An interesting observation has been made by Shull and Lbwdin(H.
23, 1958, Revs.
Mod. Phys.
Ilfl, 966 (1964).
Shull and P. 0. LtJwdin, J. Chem. Phys., & 1035 (1956) in the helium case; the SCF atomic orbital was not suitable as a starting point for the formation of split orbitals.
7. C. Edmiston and K. Ruedenberg, J. Chem. Phys., 00, 8.
2. Gershgorin and R. Pauncz,
9. R. Pauncz and
000 (this issue).
To be published.
2 . Silberman, To be published.
10. D. M. Hirst and J. W. Linnett, Proc. Chem. SOC., (1961) 427. -11. D. M. Hirst and J. W. Linnett, J. Chem. Phys., 00, 000 (this issue).
2, 1495 (1951).
12.
C. R. Mueller and H. Eyring, J. Chem. Phys.,
13.
C. A. Coulson and I. Fischer, Phil. Mag.,
14.
P, B. Empedocles and J. W. Linnett, Proc. Roy.
40,386
(1949).
SOC.,
166 (196k). e,
15. N. Epstein, (private communication).
8,
151 (1964).
16.
D. P. Chong and J. W. Linnett, Mol. Phys.,
17.
J. W. Linnett, The Electronic Structure of Molecules, A New Approach, Wiley, New York,
18.
R. &Weeny,
1964.
Proc. Roy. SOC., A223, 306 ( 1954).
19. P. 0. Ltfwdin, J. Appl. Phys., Suppl. No. 1,
251 (1962).
20.
R. Pauncz, J. de Heer, P. 0. Lbdin, J. Chem. Phys., ( 1962)
21.
J. L. Calais, Arkiv f. Fysik, 00, 000 (1965).
5, 2247~2257